Concave Square with Straight Sides
This can be accomplished by using proportional editing.


Add a cube.





Switch to edit mode (Tab) and subdivide it (F3 > Subdivide). Use an odd number of subdivisions in order to have a single vertex at the center of the cube face. Use a low number of subdivisions, e.g. 5.







Select that single vertex at the center and enable proportional editing (O) and set the falloff type to Sphere.





Move the vertex down on the z-axis (G + Z). If you're not using the default dimensions of the cube you may have to adjust the radius using the scroll wheel of your mouse. The outer edges of the cube shouldn't be within the radius of the proportional editing.